Suslin rigidity theorem

Remarks

The idea of the proof starts with a smooth curve $C$ over $k$ and two $k$-points $x, y$ of $C$. The two specialization maps from the mod $\ell$ K-theory of $C$ to that of $k$ agree, because the relevant part of the Picard group of $C$ is $\ell$-divisible. The field $L$ is a union of finitely generated $k$-algebras, each of which has $k$-points. Specialization at such points shows that the map is injective, and a further use of the same rigidity property shows that it is surjective.

At work: let $k = \overline{\F}_p$ and $\ell$ prime to $p$. Passing to the colimit in Quillen's computation for finite fields gives $K_n(\overline{\F}_p; \Z/\ell) \cong \Z/\ell$ for even $n \ge 0$ and $0$ for odd $n$. By the theorem, the same holds for every algebraically closed field of characteristic $p$. Suslin also showed that the same answer holds in characteristic $0$, where it agrees with the mod $\ell$ homotopy of the space $\Z \times BU$ of topological K-theory.

The hypothesis that both fields are algebraically closed is necessary. For $\Q \subseteq \C$ and $\ell = 2$, $K_1(\Q; \Z/2) = \Q^\times/(\Q^\times)^2$ is infinite, while $K_1(\C; \Z/2) = 0$ because every complex number is a square. The coefficients are also needed: $K_1(\overline{\Q}) = \overline{\Q}^\times \to \C^\times = K_1(\C)$ is not surjective, since $\overline{\Q}$ is countable.

Going deeper

For a student

Rigidity is what makes mod $\ell$ K-theory computable: it reduces algebraically closed fields to the algebraic closure of the prime field. It is the basis for comparing algebraic K-theory with étale cohomology and topological K-theory. Gabber's rigidity theorem extends the idea from field extensions to Henselian local rings and their residue fields.